--- title: Max Teleportation Fidelities url: https://www.emergentmind.com/topics/average-of-maximum-teleportation-fidelities type: topic --- # Max Teleportation Fidelities Average of maximum teleportation fidelities denotes a family of optimization-and-averaging performance measures used to characterize teleportation protocols and quantum networks. In network theory, the phrase is used literally for the average, over source-target pairs, of the largest teleportation fidelity achievable for each pair under an allowed entanglement-distribution scheme. In much of the protocol literature, the nearest object is instead the maximum average teleportation fidelity: the largest input-averaged fidelity attainable for a fixed resource after optimizing over correction unitaries, measurements, gains, or controller settings. The common structure is an optimization followed by an averaging step, but the optimized variable and the averaging ensemble differ across settings [2409.20304, 2606.18743, 2511.21079]. ## 1. Definitions and terminological scope For standard teleportation of a pure input state \(|\phi\rangle\), the single-shot fidelity is the input-output overlap \(f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle\), and the average teleportation fidelity is the Haar average \[ F=\int d\phi\, f(\phi), \] with normalized Haar measure \(\int d\phi=1\). Several papers augment this first moment with the fidelity deviation \[ D=\sqrt{\int d\phi\, f(\phi)^2-F^2}, \] which quantifies the spread of fidelities over the input ensemble rather than the mean alone [2511.21079, 1801.06115, 2102.05396]. Within this protocol-level usage, the relevant “maximum” is usually not \(\int d\phi\,\max f(\phi)\). Rather, it is the largest value of the average \(F\) obtainable after optimizing over admissible protocol parameters. For isotropic shared resources, this optimization is typically over Bob’s correction structure \(\{\hat V_\alpha\}\), equivalently over the composed unitaries \(\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger\) [2511.21079]. In two-qubit analyses it is often an optimization over local-unitary strategies within the standard protocol [1906.01394]. In continuous-variable teleportation it is the gain optimization \(\max_g \mathcal F\) [2203.06684]. In controlled teleportation it is an average over branchwise maximal fidelities, followed by an optimization over Charlie’s measurement basis [2606.10826]. A further notational ambiguity comes from the Horodecki-type relation \[ \bar f_{\max}=\frac{dF_{\max}+1}{d+1}, \] used in qudit analyses where \(F_{\max}\) denotes the maximal overlap with a maximally entangled state, rather than the averaged teleportation fidelity itself [1305.3443]. By contrast, network papers use the exact phrase “average of maximum teleportation fidelity” for a graph-level quantity, \[ F^{\rm max}_{\rm avg}(\rho)=\langle F^{\rm max}_{ST,\mathcal P_{\max}}(\rho)\rangle_{ST}, \] namely the average over node pairs of the best end-to-end teleportation fidelity permitted by the network and protocol class [2409.20304, 2606.18743]. ## 2. Maximum average fidelity for fixed resources and protocol classes For \(d\)-dimensional teleportation with isotropic shared resource \[ \hat\rho_{\mathrm{iso}(p)}=p\,|\Psi_0\rangle\langle\Psi_0|+\frac{1-p}{d^2}\hat{\openone}_{d^2}, \] the average teleportation fidelity takes the closed form \[ F=\frac{p}{d^3(d+1)}\sum_{\alpha=0}^{d^2-1}\bigl(\operatorname{Tr}\hat X_\alpha\bigr)^2+\frac{d+1-p}{d(d+1)}. \] Its maximum is achieved when all composed corrections are identity, \[ \hat X_\alpha=\hat{\openone}_d \quad \forall\alpha, \] equivalently \(\hat V_\alpha=\hat U_\alpha\), and then \[ F_{\max}=p+\frac{1-p}{d}. \] The corresponding classical benchmark is \[ F_{\mathrm{cl}}=\frac{2}{d+1}, \] and for isotropic states one has \[ p>\frac{1}{d+1}\Longleftrightarrow F_{\max}>\frac{2}{d+1}. \] Thus the optimum average fidelity is fixed exactly by the isotropic visibility parameter and the dimension [2511.21079]. In the qubit Werner specialization, the same structure yields \[ F_{\max}=\frac{1+p}{2}, \] with the optimum attained by matching Bob’s correction unitaries to Alice’s Bell-basis labels, \(\hat V_\alpha=\hat U_\alpha\). The classical threshold is \(2/3\), so useful qubit teleportation requires \(p>1/3\) [1801.06115, 2102.05396]. For an arbitrary two-qubit resource state \(\rho\), the maximal average fidelity achievable within the standard protocol and local-unitary strategies is \[ F_\rho=\frac{2\mathcal F_\rho+1}{3}, \] where \(\mathcal F_\rho\) is the fully entangled fraction. In the canonical form with diagonal correlation matrix \(T\), the closed expression is \[ F_\rho= \begin{cases} \frac13\left(1+\sum_{i=1}^3 |t_{ii}|\right), & \det T\le 0,\\[1.2ex] \frac13\left[1+\max_{ijk}\left(|t_{ii}|+|t_{jj}|-|t_{kk}|\right)\right], & \det T>0. \end{cases} \] The state is useful for teleportation iff \(F_\rho>2/3\) [1906.01394]. Qudit teleportation with noisy classical communication provides another realization of the same optimization logic. In the Bell-measurement protocol considered there, the average fidelity is \[ f_{\textrm{avg}}=\frac{d\bar p_{00}+1}{d+1}, \] where \(\bar p_{00}\) is the average probability that the Bell outcome is transmitted correctly. The same paper connects this to the Horodecki form \[ \bar f_{\max}=\frac{dF_{\max}+1}{d+1}, \] with \(F_{\max}=p_{00}\) for the induced noisy resource. The classical threshold remains \(2/(d+1)\), and beating it requires \(\bar p_{00}\ge 1/d\) [1305.3443]. ## 3. Fidelity deviation, full fidelity distributions, and universality The protocol literature increasingly treats the average fidelity as insufficient by itself. For isotropic \(d\)-dimensional teleportation, the deviation obeys the tight bound \[ D\le \sqrt{\frac{2}{d(d+3)}\bigl(F_{\max}-F\bigr)}. \] Hence \(F=F_{\max}\) forces \(D=0\), and the \((F,D)\) plane acquires a witness geometry. Writing \[ s_d=\sqrt{\frac{2}{d(d+3)}}, \] the teleportation-advantage and Bell-nonlocality witnesses become \[ F+\frac{D}{s_d}>\frac{2}{d+1}, \qquad F+\frac{D}{s_d}>F_{\mathrm{BV}}^{\max}(d). \] The identical slope and different intercepts separate the regions of teleportation advantage and CGLMP-inequality violation [2511.21079]. For optimal teleportation with an arbitrary two-qubit state, the fidelity deviation can itself be computed exactly. In the useful sector \(\det T<0\), the universality condition is \[ \Delta_\rho=0 \Longleftrightarrow |t_{11}|=|t_{22}|=|t_{33}|. \] Moreover, for every prescribed maximal average fidelity \(F_\rho>2/3\), there exist states with \(\Delta_\rho=0\). This identifies a large class of states that are simultaneously useful and dispersion-free [1906.01394]. A stronger statistical extension replaces the first two moments by the full probability density function of actual teleportation fidelities. For single-qubit teleportation, the actual fidelity is treated as a random variable over Haar-random inputs and protocol outcomes, and the general PDF is \[ f(F)=\sum_j \frac{1}{4\pi}\int d\mathcal S_j\, \frac{p_{j|\vec x}}{|\nabla F_j(\vec x)|}. \] For the optimal classical measure-and-prepare protocol this reduces to \[ f(F)=2F, \] with \(\langle F\rangle=2/3\) and support on the full interval \([0,1]\). The paper shows that protocols with identical average fidelities can have markedly different distributions, and that local amplitude damping can introduce asymmetries invisible at the level of \(\langle F\rangle\) alone [2510.13600]. This suggests that the “maximum” relevant to teleportation quality can refer not only to an optimized mean, but also to support endpoints, upper-tail probabilities, or prior-weighted certification scores. In that sense, the average of maxima and the maximum of averages are only two members of a broader family of fidelity-sensitive diagnostics [2510.13600]. ## 4. Benchmarks beyond Haar-uniform pure inputs When the input ensemble is not Haar-uniform over pure qubits, the benchmark itself changes. For a von Mises–Fisher prior on the Bloch sphere, \[ P(\theta,\phi)=\xi e^{\kappa\cos\theta}, \qquad \xi=\frac{\kappa}{4\pi\sinh\kappa}, \] the highest mean fidelity achievable without entanglement depends on the concentration parameter \(\kappa\). For the single-qubit projective strategy optimized over measurement axis and Bayesian reconstruction, the exact benchmark is \[ \langle F_{\max}^{(1)} \rangle =\frac12\left[1+\left(\coth\kappa-\frac{1}{\kappa}\right)\sqrt{1+\frac{1}{\kappa^2}}\right], \] attained at a measurement axis perpendicular to the prior peak, \(\theta_0=\pi/2\). As \(\kappa\to 0\), this reduces to \(2/3\); as \(\kappa\to\infty\), it approaches \(1\). Hence nonuniform prior information can raise the classical no-entanglement threshold far above the uniform-input value [2605.24861]. A different generalization concerns isotropic distributions of mixed input states. For single-qubit mixed inputs drawn isotropically from the Bloch ball, the standard teleportation protocol with arbitrary resource state admits a maximal average fidelity \[ \overline{F}_{\max} =\frac12\left[ 1+\frac{8\pi\alpha}{3}\left(2\mathcal F(\rho)-\frac12\right) +\int_{\mathcal B} d^3\mathbf t\, f(t)\sqrt{1-t^2}\sum_i p_i\sqrt{1-\mathbf t_{C|i}^2} \right]. \] For Werner resources this becomes \[ \overline F_{\max} =\frac12\left( 1+4\pi\alpha\,p +4\pi\int_0^1 dt\, t^2 f(t)\sqrt{(1-t^2)(1-p^2 t^2)} \right), \] and for fixed-purity inputs of Bloch radius \(x\), \[ \overline F_{\max,x} =\frac12\left(1+px^2+\sqrt{(1-x^2)(1-p^2x^2)}\right). \] In this mixed-input setting, the paper shows that separable Werner states and even Bell-diagonal classical-quantum states can exceed the corresponding classical benchmark, whereas the computational basis on Alice’s side never does so for any resource state [2109.14479]. Noisy X-state channels exhibit a different threshold structure. For the standard qubit protocol, \[ f_X=\frac{2}{3}+\frac{1}{3}\left[2\rho_{14}-(\rho_{22}+\rho_{33})\right] =\frac{2}{3}+\frac{C_{14}-(\sqrt{\rho_{22}}-\sqrt{\rho_{33}})^2}{3}. \] Thus entanglement is necessary but not sufficient for quantum advantage; the threshold concurrence is \[ C_{X,\mathrm{th}}=(\sqrt{\rho_{22}}-\sqrt{\rho_{33}})^2. \] After unambiguous state extraction, the total average fidelity decreases, \(f_{X,\mathrm{USE}}\le f_X\), but the successful postselected branch satisfies \(f_{X,\mathrm{USE},0}>f_X\), so fidelity can be redistributed into a higher-quality subensemble [1508.01417]. ## 5. Network-level averages of pairwise maxima In quantum-network studies, average of maximum teleportation fidelities is a literal graph-level figure of merit. For a pair \((S,T)\), one first identifies the path or entanglement-distribution strategy that yields the largest end-to-end teleportation fidelity, and then averages that optimal pairwise value over all source-target pairs: \[ F^{\rm max}_{\rm avg}(\rho) =\langle F^{\rm max}_{ST,\mathcal P_{\max}}(\rho)\rangle_{ST}. \] For loopless networks this simplifies because the path is unique, so the “maximum” is formal rather than nontrivial [2409.20304, 2508.10417]. With Werner-state links, entanglement swapping along a path \(\mathcal P\) yields \[ F^{\rm max}_{ST,\mathcal P}(\rho_{\rm wer}) =\frac{1+\prod_{i\in\mathcal P}p_i}{2}. \] In a homogeneous network with link parameter \(p\) and path length \(\ell\), this becomes \[ F^{\rm tel}_\ell=\frac{1+p^\ell}{2}. \] The classical benchmark remains \(2/3\), so for a path of length \(\ell\) quantum advantage requires \(p^\ell>1/3\) [2409.20304, 2606.18743]. For loopless repeater networks, this metric sharply distinguishes topologies. In stars and chains with identical links, the star maximizes the network average while the chain minimizes it. In the large-\(N\) limit, \[ F^{\rm max}_{\rm avg}\to \frac{1+p^2}{2} \quad\text{for the star}, \qquad F^{\rm max}_{\rm avg}\to \frac12 \quad\text{for the chain}. \] Hence a large star retains quantum advantage iff \(p>1/\sqrt3\), whereas large chains lose it on average [2409.20304]. The metric becomes more operationally flexible when multiple paths are allowed and sequential purification is permitted. In that setting, the average network fidelity is evaluated after producing several end-to-end states and purifying them with the Deutsch protocol, \[ \mathcal P_D(f_1,f_2)=\frac{10f_1f_2-f_1-f_2+1}{8f_1f_2-2f_1-2f_2+5}. \] Because purification is commutative but not associative, the ordering matters. For homogeneous networks, the best ordering is Shortest Path Last (SPL), and purification can substantially enhance average teleportation fidelity in highly connected topologies such as complete graphs, triangular lattices, square lattices, and Erdős–Rényi networks, while offering little benefit in large rings [2606.18743]. Binary-tree repeater networks provide a finer topology comparison. For directed asymmetric, directed symmetric, undirected asymmetric, and undirected symmetric binary trees, closed analytic expressions can be written as \[ F^{\text{tel}_{\text{avg}}=\frac{\sum_r l_d^{(r)}\mathcal F_r}{\sum_r l_d^{(r)}}, \qquad \mathcal F_r=\frac{1+p^r}{2}, \] where \(l_d^{(r)}\) counts allowed length-\(r\) paths. The directed symmetric binary tree (DSBT) is the best of the four classes. For \(N=15\), the threshold link qualities for \(F^{\text{tel}_{\text{avg}}}>2/3\) are \[ 0.638\ \text{(DABT)},\quad 0.511\ \text{(DSBT)},\quad 0.699\ \text{(UABT)},\quad 0.698\ \text{(USBT)}, \] and for \(N=127\) they are \[ 0.931,\quad 0.666,\quad 0.936,\quad 0.868, \] respectively. All four topologies satisfy \[ \lim_{N\to\infty}F^{\text{tel}_{\text{avg}}}=\frac12, \] but DSBT approaches that limit slowest and is therefore the most favorable finite-size binary-tree architecture under this metric [2508.10417]. ## 6. Generalizations, experiments, and persistent ambiguities Controlled teleportation introduces a nested optimization structure that is especially close to the literal phrase “average of maximum teleportation fidelities.” After Charlie’s two measurement outcomes, the paper defines the branchwise maximal fidelities \[ F_{m\max}=\frac13+\frac23\max_i\langle\Phi_i|\rho_{mBC}|\Phi_i\rangle, \] and then averages them: \[ F_{\max}=p_1F_{1\max}+p_2F_{2\max}. \] A further optimization over Charlie’s measurement angles \((\theta,\varphi)\) gives the optimal average fidelity \(\mathcal F_{\max}\). Under generalized noisy channels interpolating between amplitude damping and dephasing, the dependence on the evolution parameter need not be monotonic: when only Charlie’s qubit is noisy and \(a_1>a_0\), the optimal average fidelity first decreases and then increases, whereas for \(a_1